It essentially takes the place of the particle ga or wa after the subject.
これこそ私の欲しい本です。
Kore koso watashi no hoshii hon desu.
This is the very book that I want.
私こそお礼お言わなければなりません。
Watashi koso orei o iwanakereba narimasen.
I(not you) should expressing thanks.